Example:
What is inf, sup, lim inf and lim sup for j}.gif)
?
Clearly, the infimum of the sequence is -1, and the supremum is
+1. To find lim inf and lim sup, we will first
find the sequence of numbers
Aj and
Bj mentioned in the definition.
Let's find the numbers
Aj =
inf{aj, aj + 1, aj + 2, ...}
for this sequence {-1, 1, -1, 1, ...}.
- A1 = inf{-1, 1, -1, 1, ...} = -1
- A2= inf{1, -1, 1, -1, ... } = -1
and so on. Therefore, it is clear that
- lim inf = -1
Similarly, we find the numbers
Bj =
sup{aj, aj + 1, Aj + 2, ...}
- B1 = sup{-1, 1, -1, 1, ...} = 1
- B2 = sup{1, -1, 1, -1, ...} = 1
and so on. Therefore, it is clear that
- lim sup= 1
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